concrete pavement slab under blast loads · the load location on the concrete slab is also depicted...

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International Journal of Impact Engineering 32 (2006) 1248–1266

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Concrete pavement slab under blast loads

Bibiana Marıa Luccionia,�, Mariela Luegeb

aStructures Institute, National University of Tucuman, CONICET, Juan B. Teran 375, 4107 Yerba Buena,

Tucuman, ArgentinabStructures Institute, National University of Tucuman, CONICET, Av. Roca 1800, 4000 S.M. de Tucuman, Argentina

Received 22 March 2004; received in revised form 7 August 2004; accepted 27 September 2004

Available online 26 November 2004

Abstract

The main objective of this paper is the analysis of the behaviour of concrete pavements subjected to blastloads produced by the detonation of high explosive charges above them. This subject is of particular interestin terrorist attacks in cities, since important conclusions about the location and magnitude of the explosivecharge can be drawn from simple observation of the pavement damage. Experimental results for a concreteslab lying on the ground and subjected to blast loads are presented. The slab was lying on the ground and wastested with different amounts of explosive suspended in air over it. Numerical results are compared with thoseobtained with a simplified plastic limit analysis and those obtained with two types of numerical simulationsperformed with two different codes. Some conclusions about the effect of the blast load on the concrete slaband about different tools available for the analysis of this type of problem are stated in the paper.r 2004 Elsevier Ltd. All rights reserved.

Keywords: Concrete slabs; Blast load; Experimental analysis; Limit analysis; Numerical analysis

1. Introduction

Due to different accidental or intentional events in connection with important structures allover the world, blast loads have received considerable attention in recent years [1–3]. The activityrelated to terrorist attacks has increased and, unfortunately, the present tendency suggests that itwill be even larger in the future. This paper is concerned with the effect of dynamic loading

see front matter r 2004 Elsevier Ltd. All rights reserved.

ijimpeng.2004.09.005

ding author. Tel.: +54 381 4364087; fax: +54 381 4364087.

ress: [email protected] (B.M. Luccioni).

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B.M. Luccioni, M. Luege / International Journal of Impact Engineering 32 (2006) 1248–1266 1249

produced by the detonation of high explosives on concrete pavements, a situation likely in asignificant number of terrorist attacks.When the attack has already occurred, a very important issue is the determination of the

location of the explosion and the mass of explosive used. A useful tool to achieve this objective isthe evidence of the crater generated by the explosion in the pavement.A blast wave originating from a closed or free explosive detonation behaves, when interacting

with structures, as a short duration dynamic load. In recent years, studies have shown that suchloads with short duration and high magnitude influence significantly the response of the structureand can modify substantially the expected material behaviour [4–9].Much research has been carried out in recent years concerning the behaviour of structural

elements and materials under blast loads. Experimental results about the behaviour of steel [10,11],concrete [12,13] and fibre reinforced [14] panels subjected to blast loads can be found in References.In the literature, beams, slabs and shells under blast loads are mostly studied with limit analysis

theory which assumes a rigid-plastic behaviour for the material [12,15]. Yi [12] tested concreteslabs under blast loads and studied the behaviour using a non-linear dynamic analysis. For close-in explosions the problem was solved by an approximate procedure. First a check is performed todetermine if the zone just below the load has disintegrated or not. If it has disintegrated, the levelof damage is estimated. The tension failure of the other side of the plate is also studied with theaid of elastic theory. For the assessment of the dynamic displacement of the plate centre, a onedegree of freedom model is used.On the other hand, with the rapid development of computer hardware over the last decades, it

has become possible to make detailed numerical simulations of blast loads on personal computers,significantly increasing the availability of these methods.Although there are still many uncertainties, material behaviour under blast loads has been

widely studied experimentally [16–19] and many sophisticated numerical models have beenproposed, especially for steel and concrete [4,6,9,20–23]. The behaviour of concrete under blastloads is characterized by a different response in tension and compression, the progressivedegradation of elastic properties accompanied by increasing permanent strains and thedependence of the strength, stiffness and fracture energy on strain rate. This last feature hasbeen normally taken into account through viscoplastic models or rate-dependent damage models.[4–9]. These models have been included in different computer programs [4,23,24], which can beused for the analysis of the blast behaviour of structural elements and small structures andvalidated with available experimental results.In this work the experimental results for a concrete plate lying on the ground, subjected to different

blast loads, are compared with the results of a simplified limit analysis and numerical results obtainedwith the ABAQUS/Explicit [25] program and AUTODYN 2D and 3D [26]. The dimensions of thedamage zone are also compared with those obtained by Yi [12] and with the dimensions of the cratersthat would have been produced by the explosions acting directly on the ground.

2. Experimental tests

The experimental results described below belong to a series of tests carried out by the StructuresInstitute of the National University of Tucuman. The complete series of tests consisted of the

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B.M. Luccioni, M. Luege / International Journal of Impact Engineering 32 (2006) 1248–12661250

detonation of explosive charges lying on the ground and elevated above the ground [10,11] andabove a concrete plate lying on the ground. Pressure loads were registered at different distances,together with the size of the craters formed in the soil, the response of two metallic panels and thecracking and damage patterns of the concrete slab. In order to measure the overpressuregenerated by shock waves, pressure sensors were used. In addition, accelerometers were used tomeasure the dynamic response of the steel plates. A dynamic strain amplifier amplified the signalgenerated by the accelerometers. A data acquisition board was mounted on a notebook computerin order to record and process the signals by means of data acquisition software.This paper is focused on the behaviour of the concrete slab. The geometrical configuration of

the reinforced concrete slab lying on the ground is shown in Fig. 1. The reinforcement consisted of4.2mm diameter bars spaced 150mm in both directions. The average compressive strength of theconcrete (25MPa) was obtained from compression tests at 28 days of a series of cylindical proofsamples cast with the same concrete as the slab.The soil was soft-OL type (Unified System for the Classification of Soils), i.e. an organic soil

without fine particles (more than 50% passing through sieve #200) with a liquid limit of less than50 and a plastic limit of less than 4.The load location on the concrete slab is also depicted in Fig. 1. Spherical explosive charges of 5

and 12.5 kg of Gelamon VF80 were employed placed at 0.50m height above the top surface of theslab as shown in Fig. 2. The nominal TNT equivalence factor for Gelamon VF80 is 0.8. Table 1summarizes the sequence of the applied loads and their locations.The cracking patterns registered after the experimental tests are shown in Figs. 3 and 4. The

charge of 5 kg Gelamon produced the crushing of concrete in a circular zone of about 250mmdiameter, while the diameter of the concrete crushing zone was about 300mm for the 12.5 kgcharge. The first test produced a fracture of the slab parallel to the short side. As a result, for thefollowing detonations, the original slab behaved as two square independent slabs. Furthermore,circumferential cracks could be found around the crushing zone in both cases. Such cracks definedtwo circumferences, the first one with a diameter of about 1.0m and the second one coincident

Fig. 1. Concrete slab dimensions and charge positions.

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Fig. 2. Placement of the explosive charge suspended above the slab.

Table 1

Explosion sequence

Test Position Height (m) Charge (kg of Gelamon)

1 A 0.5 5

2 A 0.5 5

3 B 0.5 12.5


with the border of the slab and both centred on the point on the slab directly beneath the charge.Radial cracks could be seen quite clearly, in particular the ones parallel to the sides of the slab.The relative maximum vertical displacements were found to be located directly beneath the two

charge locations A and B. The values registered were 23mm for position A, and 75mm forposition B.The relative diameter of the crushing zone D to the height of the charge h is shown in Fig. 5a as

a function of the inverse of the scaled distance W 1=3=h: The results obtained by Yu [12] forexplosions in direct contact with a concrete plate have also been included in Fig. 5a together withthe results corresponding to craters in soils [27]. It can be observed that, although the crushing

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Fig. 3. Cracking patterns (a) after test 1, (b) after test 2, (c) after test 3.


zone in concrete is always smaller than the corresponding crater in soil, the results follow the sametrend. Fig. 5b shows the results of craters on concrete slabs and a curve corresponding to andexponential approximation of the relation presented in Eq. (1). This equation can be used eitherfor the estimation of explosive charges from crater diameters or crater dimensions from explosivecharge, depending on what the purpose of the analysis is

lnð3:63D=hÞ ¼ 0:1838ðW 1=3=hÞ: (1)

3. Limit analysis

The limit analysis of plates with different geometry and boundary conditions subjected todynamic loads is well documented in the literature [15].The equation of motion of a circular rigid perfectly plastic plate, of radius a, simply supported

on its boundary and subjected to uniform distributed pressure, as indicated in Fig. 6a, is given byLubliner [15]

@ rMrð Þ

@r�My ¼

Z r

0

pþ m@2w

@t2

� �r dr; (2)

where m is mass per unit area, p is pressure, Mr is radial moment, My is circumferential moment, wis vertical displacement and r is radius. The generalized Tresca yield criterion with ultimatebending moment Mu is assumed (see Fig. 6b). All the plate is supposed to be in regime BC

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Fig. 4. Damage of the slab after the tests.


(Fig. 6b): My ¼Mu; whereas for the centre of the plate Mr ¼My ¼Mu (Point C in Fig. 6b).Consequently, it can be assumed that the plate has a conic deformation (Fig. 6c) described by thefollowing equation:

wðr; tÞ ¼ �DðtÞ 1�r

a

h i; (3)

where the parameter D(t) represents the deflection of plate centre at time t.Eq. (3) together with Eq. (2), My ¼Mu and the boundary condition Mrða; tÞ ¼ 0; gives,

Mr ¼MU �r2

6p� ðp� pU Þ 2�

r

a

� �h i; (4)

where pU ¼ 6MU=a2 is the static limit pressure.Eq. (4) is valid for pp2 pU : For the case p42 pU ; the plate configuration is a truncated cone,

with a circular hinge of radius r0 separating the external conic zone from the internal one, seeFig. 6d. The latter corresponds to the point B of the Tresca criterion ðMr ¼My ¼MU Þ and theequations of motion that result are

m@2D@t2¼ p for rpr0; (5a)

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0.1

1

10

100

1 10 100

W1/3/h [kg1/3/m]

D/h

EMRTC on the ground

Above the ground [27]

Above 150 mm Concrete slab

On 220mm Concrete slab [12]

On 390 mm concrete slab [12]

ln (3.63 D/h)=0.1838 W1/3/h

0.1

1

10

100

0 5 10 15 20

W1/3/h [kg1/3/m]

D/h

Experimentalresults

Exponentialapprox. Eq. (1)

On the ground [27]

(a)

(b)

WhD

Fig. 5. Variation of the relative diameter of the crushing zone D/h as a function of the inverse of the scaled distance

W 1=3=h (a) comparison with the dimensions of craters in soils, (b) exponential approximation.


@ðrMrÞ

@r¼MU �

pð2r3 � 3r0r2 þ r30Þ

6ða� r0Þfor r4r0: (5b)

Integrating Eq. (5b) with the initial condition Mr ¼MU at r ¼ r0; the following cubic equationis obtained for r0:

1�r0

a

� �21þ

r0

a

� �¼

2pU

p: (6)

Notice that the substitution of r0 ¼ 0 in Eq. (6) gives p ¼ 2 pU :

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MU

MU

M�

Mr

A

BC

�(t)

a

p

r o

�(t)

(a) (b)

(c) (d)

Fig. 6. Circular rigid perfectly plastic plate (a) Undeformed shape, (b) Tresca Criterion, (c) Ultimate configuration for

pp2 pu and (d) Ultimate configuration for p42 pu:


According to Lubliner [15], an estimation of the upper bound for the limit load of a clampedplate of arbitrary shape can be obtained from the previous results. It could be considered that thelargest circle that can fit into the plate is a hinge. The material inside this circle collapses like aclamped circular plate and the material outside it remains rigid.An approximate solution for the tested slab can be obtained carrying out ultimate analysis,

making the assumption that the soil under the slab does not alter its collapse mechanism.Following Lubliner [15], a circular clamped plate of radius a ¼ 0:75m is considered. As thereinforcement is placed at the back face of the plate, the negative ultimate moment is almost zeroand the circular plate can be supposed to be simply supported at its border. Therefore, Eq. (6) canbe used to estimate an upper limit for the ultimate load or an upper limit to r0. The ultimatepositive moment is defined by the yielding of the reinforcement. The external cone radius r0, canbe obtained from Eq. (6), with pU ¼ 6MU=a2 ¼ 9:28 MPa: The results obtained can besummarized as follows:

�
Position A: 4 kg TNT at 0.5m height (p ¼ 63:8MPa): r0 ¼ 0:45m; � Position B: 10 kg TNT at 0.5m height (p ¼ 151:1MPa): r0 ¼ 0:55m:
It can be observed that the radius r0, calculated using the classical theory, is almost coincidentwith the radius of the smallest circumferences that are shown in Fig. 3. It can be assumed that theslab responded as a simply supported circular plate whose boundary was the externalcircumferential crack, almost coincident to the border of the slab. As the slab was slightly

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reinforced, the reinforcing steel was not enough to spread the cracking effect, like in thecontinuum model used, and appreciable discrete cracks were formed. Moreover, as the externalcrack was not exactly circular but parallel to the sides of the slabs radial cracks can be foundwhere My ¼MU and where the plastic deformation was concentrated.

4. Numerical simulation

The numerical analysis has been carried out with two different codes: the finite elementprogram ABAQUS/Explicit 5.8-18 [25] and the hydrocode AUTODYN v. 4.3 [26].

4.1. Finite element program [25]

The finite element mesh of the whole system is given in Fig. 7. The slab was modelled with 1000C3D8R tri-dimensional elements arranged in five layers of equal height. The reinforcement wasmodelled by adding two layers of steel (REBAR) placed between the first and second layer and inboth directions 1 and 2.For the soil, a combination of C3D8R tri-dimensional elements and CIN3D8 semi-infinite tri-

dimensional elements were used, so that the soil could be approximately modelled as a semi-infinite space. The semi-infinite elements represent smooth boundaries through which energy canpass without being absorbed or reflected.Materials were characterized using simple constitutive models, with the idea of capturing the

global response of the system with the use of few observable parameters.

Fig. 7. Finite element mesh (ABAQUS CAE 2-5).

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An elasto-plastic material model with properties dependent on the strain rate has been used forthe concrete. The material model is the extended Drucker–Prager with the parameters reported inTable 2.Steel was characterized with an elasto-perfectly plastic model with the properties presented in

Table 3.The elasto-plastic material model used for the soil was the modified Drucker–Prager/Cap. All

the model material properties were estimated on the basis of the type of soil (due to the lack ofexperimental data) and are presented in Table 4.The detonation of an explosive charge generates a pressure wave in the air. The evolution of the

pressure is characterized by a strong increment of pressure at the wave front followed by anexponential fall. The peak value of the overpressure depends on the mass of explosive and on thedistance to the detonation point.To define the blast load the ABAQUS/Explicit user subroutine VDLOAD was used [25]. This

subroutine allows the user to define the load as a function of position and time. The blast load wasmodelled as pressure acting on the exposed face of the slab. The value of the pressure is a spaceand time dependent function which must take into account the time taken by the overpressurepeak to meet the slab, the decrease of the peak pressure value with the distance and its exponentialreduction with time [28]. In order to define those aspects the following empirical expressions wereused [28]:

pðtÞ ¼ 0 toto

pðtÞ ¼ ðpr � psÞ 1�t� to

t� t0 � to

� �þ ps 1�

t� to

T s

� �exp �

b t� toð Þ

T s

� �tootol

pðtÞ ¼ ps 1�t� to

T s

� �exp �

b t� toð Þ

T s

� �t4to þ t0

(7)

ps ¼1:4072

Zþ

0:5540

Z2�

0:0357

Z3þ0:000625

Z4½MPa� 0:05pZp0:3

ps ¼0:6194

Z�

0:0326

Z2þ

0:2132

Z3½MPa�

0:3pZp1:0

ps ¼0:0662

Zþ

0:405

Z2þ0:3288

Z3½MPa�

1:0pZp10

(8)

Table 2

Concrete mechanical properties

Density: 2400 kg/m3

Elastic modulus: 2.5� 104MPa

Poisson’s ratio: 0.15

Compressive elastic limit: 18MPa

Compressive strength: 25MPa

Internal friction angle: 301

Dilatancy angle: 201

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Table 4

Soil mechanical properties

Density: 1200 kg/m3

Elastic modulus: 100MPa


Cohesion: 0.11MPa

Internal friction angle: 201

Table 3

Steel mechanical properties

Density: 7800 kg/m3




Z ¼R

W 1=3; pr ¼ 2ps

7po þ 4ps

7po þ ps

� �;

t0 ¼ 3S=U s; U s ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi6ps þ po

7po

sao; to ¼ R=U s;

ð9Þ

where R is the distance to the charge, W is the mass of the explosive [kg of TNT], Z is the scaleddistance, p the pressure acting on the slab, ps is the overpressure at the wave front, pr is thereflected pressure, po is the atmospheric pressure, t is the time, to is the time when the overpressuremeets the structure, Ts is the duration of the positive overpressure phase that can be obtainedfrom tables as a function of Z, t0 is the time of incidence of the reflected pressure, b is the waveshape parameter that depends on ps, Us is the velocity of the wave front, ao is sound velocity atambient conditions, S is half the slab short side length.Normal reflection and validity of Rankine–Hugoniot (R–H) relations were assumed. These

simplifying assumptions were made in order to obtain simple analytical expressions that could bewritten in a user subroutine. It should be noted that these constitute great simplifications takinginto account the proximity of the blast load to the slab. This fact shows one of the majordrawbacks of this type of numerical analysis where the blast load has to be approximated since acomplete coupled analysis cannot be performed.To get an idea of the value of the resultant pressure magnitude and the positive phase duration,

for the 12.5 kg Gelamon charge (equivalent to 10 kg TNT), for instance, the maximum resultantpressure is 110MPa and the duration of the positive phase is around 3� 10�4 s.In order to define the load history, the blast loads were assumed to be applied at time intervals

such that the superposition of pressure waves was negligible and the response was stabilized. Thetotal duration of the load for the three tests was approximately 2.0� 10�2 s.An explicit dynamic analysis was carried out based on the central difference integration rule,

which is suitable for impulsive loads and strong discontinuous responses. The central differencemethod is conditionally stable and the stability limit depends on the highest frequency of the

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system omax [25,29]

Dtp2

omax: (10)

An approximation for this stability limit can be obtained as the shortest time interval necessaryfor p-waves to pass the elements of the mesh,

DtpminL

cd

� �; (11)

where L is the length of the element and cd the propagation velocity of p-waves.ABAQUS/Explicit program [25] allows the automatic estimation of a stable time increment.

With this purpose, the velocity of the dilatation wave is calculated in each element, for each timeincrement, assuming a hypo-elastic material model.At first, the problem was run with automatic time increment and element by element

estimation, but the response was unstable. This was due to the simplified material modelemployed for the estimation of the critical time increment.In order to obtain stability, fixed time increments of 5.4� 10�6 s were used (0.05 times the

original estimation). A total of about 7.3� 104 time increments and 90min of CPU time wererequired for the simulation of the 2� 10�2 s history.The deformation of the slab after each of the three experiments is presented in Fig. 8. It can be

seen, especially in the later tests, there is elevation of the plate corners that was also observed inexperimental tests. Points A and B suffered the highest displacements which amounted to 19 and40mm, respectively.Fig. 9 depicts the maximum stress distribution occurring during the first test on the upper and

lower face, respectively. It is interesting to notice the traction zones in coincidence with thecircumferential external cracking as revealed experimentally. Stress and strain states obtained forthe lower reinforcement indicate that the yield tension was reached in correspondence to theinternal circumferential crack, confirming the failure mechanism experimentally observed.

Fig. 8. Slab deformation (amplification factor: 20) (ABAQUS CAE 2-5) (a) test 1, (b) test 2, (c) test 3.

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4.2. Hydrocode [26]

In contrast with the analysis done with the finite element program, the complete phenomenonwas modelled in this case. New developments in integrated computer hydrocodes [30], such as theAUTODYN Software [26] used in this section, are especially suitable for the study of interactionproblems involving multiple systems of structures, fluids, and gases and highly time-dependentand non-linear problems.The analysis began with the modelling of the detonation and propagation of the pressure wave

inside the explosive and in the air in contact with the explosive. As this analysis must beperformed with much detail, it was done in a previous stage in which a spherical explosive wasmodelled with AUTODYN 2D [26,31]. Then the results of this first analysis were mapped into thethree dimensional model [26]. Starting from this point, the propagation of the blast wave in airand its interaction with the plate were simulated with AUTODYN 3D.The model was composed of the plate, the underlying soil and the air volume surrounding the

plate, as shown in Fig. 10a.A three-dimensional Euler FCT [26] (higher order Euler processor) subgrid was used for the air.

Flow out of air was allowed at all the borders. To model the air a fluid with the propertiespresented in Table 5 was used.

Fig. 9. Maximum stress contours (Pa) (test 1) (ABAQUS CAE 2-5)—(a) Upper face and (b) Lower face.

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The reinforced concrete slab was modelled with three-dimensional solid elements that weresolved with a Lagrange processor. The mesh was similar to cells that are used for the finite elementanalysis.Although reinforced concrete elements can be modelled as a combination of concrete and steel

elements joined together with the assumption of perfect bond, this type of model is prohibited foractual structures, as it requires a great number of elements. Moreover, the time step in explicitdynamic programs is directly related to the size of the elements. Elements which have dimensionssimilar to the actual reinforcement usually lead to extremely reduced time steps, making theanalysis too slow.

Fig. 10. Model for the simulation with AUTODYN—(a) Mesh, (b) Propagation of blast wave and interaction with the

concrete plate.

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Table 5

Air properties

Equation of state Ideal gas

g=1.4

r=1.225� 10�3 g/cm3

Ref. energy=0.0mJ

Press. shift=0.0 kPa

Initial conditions r=1.225� 10�3 g/cm3

Ref. energy =2.068� 105mJ/mg

Table 6

Reinforced concrete mechanical properties

Density: 2400 kg/m3



Compression strength: 25MPa

Tension strength: 4MPa


Taking into account the above considerations, an approximate material model was defined tosimulate the behaviour of reinforced concrete. The model used is a homogenized elastoplasticmaterial similar to elastoplastic models of concrete but with higher stiffness and tension strengthto take into account the influence of the reinforcement. The lower layer of the plate was filled withthat material while the upper layers were all filled with concrete with the same properties used forthe finite element analysis. The mechanical properties of the homogenized model used forreinforced concrete are shown in Table 6.The soil was also modelled with three-dimensional solid elements and solution was carried out

with a Lagrange processor. A transmit boundary condition [26] was defined at the borders tosimulate semi-infinite space. An elastoplastic model with the same mechanical properties of Table4 was used for the soil.Detonation is a type of reaction of the explosive that produces shock waves of great intensity. If

the explosive is in contact with or near a solid material, the arrival of the shock wave at the surfaceof the explosive generates intensive pressure waves that can produce the crushing or thedisintegration of the material. This shock effect is known as brisance [28]. Brisance producesdiscontinuities in the material. In order to reproduce this type of effect, the erosion model ofAUTODYN [23,24,26] was used to remove from the calculation the cells that have reached certaincriteria based on deformations. When a cell is eliminated, its mass is retained and concentrated atits nodes that begin to behave as free masses conserving their initial velocity.This erosion model represents a numerical remedial measure to counteract the great distortion

that can cause excessive deformation of the mesh. For this reason, its application to thesimulation of a physical phenomenon requires calibration with experimental results.

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The slab was simulated with different erosion models and erosion limits in order to define theappropriate erosion criteria and erosion limit. This analysis led to an erosion limit of 7.5� 10�2 ofincremental effective strain [31].To study the structural behaviour of the plate, the propagation of the blast wave and its

interaction with the plate was analysed. To do this an interaction algorithm between the Lagrange(plate) and Euler (air) processors was used [26].As an illustration, Fig. 10b represents the velocity field in the air and pressure contours on the

solids. The alteration of the blast wave produced as a result of the reflections on the plate and theground should be noted. Because of the reflections, the blast wave increased its destructive effectin the vertical direction and lost its spherical shape.In the analysis, the solid–solid interaction [26] between the plate and the soil was also taken into

account through a contact algorithm.The three explosions were simulated one after the other separated in time in order to assure that

air pressure returned to its reference value. The total real time history simulated was about 0.03 s.Increments of time of about 1� 10�7 were automatically defined by the program and a total of 1 hof CPU time was required.Fig. 11 shows the results of the numerical tests. The maximum displacements registered at

points A and B in this case are 28 and 66mm that are very close to experimental measures. Fig. 11show the crushing of concrete in the zone of the slab directly beneath the charge. The deformedshape obtained is in accordance with the failure mechanism assumed in limit analysis where theeffect of the underlying soil was supposed to be negligible.

5. Conclusions

Based on experimental results, an equation that approximately relates the crater diameter onthe pavement with the explosive charge and its height above the pavement was proposed.Due to the relatively low strength and stiffness of the underlying soil, the presence of soil does

not alter the plate behaviour. The failure shape is almost coincident with that of a circular simply-supported plate and with the diameter equal to the short side of the slab. The deformed shape ofthe plate can be approximately depicted by a truncated cone.For this case, limit analysis seems to be a simple and valuable tool to estimate the effect of a

blast load or to determine the magnitude of the explosive charge when the damage has alreadybeen done. Eq. (6) can be used to estimate the pressure producing the damage observed and then,with the aid of Eqs. (8) and (9) or charts that can be found in blast references [28], the chargeproducing that crater can be estimated.Both the finite element program and the hydrocode used approximately reproduce the

deformation and the resultant failure shape of the plate under the blast load. In both cases anexplicit dynamic analysis was required with very small time increments and a high computationalcost. Nevertheless, hydrocodes seems to be more appropriate for the simulation of the completeinteraction problem with less computational effort.Although the constitutive models used in the analysis were quite simple, they were able to

describe a similar response to the one observed experimentally. An important limitation is,however, the modelling of the discrete cracking which cannot be done with the model employed.

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Fig. 11. Deformed shape (magnification factor: 2) and concrete crushing. (a) test 1, (b) test 2, (c) test 3.


In particular, it was not possible to reproduce the division of the plate in two parts. As a result,the maximum vertical displacement obtained with the finite element model was significantly lowerthan the one measured experimentally for the third explosion using a 12.5 kg charge.More experimental data related to craters produced by explosive loads above concrete

pavements is needed. Some research effort must still be done in order to define a more suitableerosion model for concrete. Erosion models for concrete must be based on damage and failurecriteria that take into account the different response in compression and tension.

Acknowledgements

The authors are grateful to Dr. Ambrosini and Eng. Jacinto who were in charge of theexperiments, to Mr Sergio Salomon for providing the place where the experiments were carriedout and to Ms. Amelia Campos for the English revision. The financial support of CONICET andNational University of Tucuman is gratefully acknowledged.

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ARTICLE IN PRESS


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concrete pavement slab under blast loads · the load location on the concrete slab is also depicted...

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